Collision Geometry of Joint Spectra

This paper studies collision geometry for a fixed finite joint-spectrum arrangement under a one-parameter family of affine scalar projections. It separates general finite-point theorems, an exact rational nine-point census, numerical operator registration, and a conditional operator interpretation. For any fixed finite arrangement of distinct points, the collision set of the affine branches is finite, and the strict branch order is constant within each complementary chamber. When the points arise as the exact joint spectrum of commuting Hermitian operators, the collision classes are precisely the spectral layers of the corresponding affine combination. For the explicitly declared rational nine-point arrangement P9, the 36 unordered pairs split into 2 parallel, 10 interior, 15 endpoint, and 9 exterior pairs. The interior collision parameters on the unit interval are 2/7, 2/5, 1/2, 2/3, and 4/5. The parameter 2/3 is the unique interior point with maximal layer-count drop. In the declared 228-dimensional complex128 Rubik realization, the quarter-turn and half-turn averages register nine numerical joint clusters against P9. The coordinate-matched labels, ranks, and projectors are stable across the tested clustering tolerances, and the maximum raw coordinate discrepancy is below 10 to the power of minus 15. These residuals support a numerical registration but do not prove exact commutation, exact rational joint eigenvalues, or exact primitive idempotents. Conditional on exact registration, the six averaging layers are the collision quotient of P9 at parameter 2/3. The paper keeps the point arrangement fixed and varies only the affine projection. Parameter-dependent spectral charts, moving arrangements, maximal commuting refinements, and information-flow preservation remain research questions. Collision adjacency is also kept distinct from direct transport and projected matrix composition. Source repository: https://github.com/dooven-prime/rime-lite Paper IV of the Representation-Induced Mechanics and Evolution (RIME) program. Independent researcher; no institutional affiliation.